Israeli television reported on Friday that Shaul Mofaz, the defence minister, told Abbas that unless a crackdown on militants begins by Sunday, Israel will redesignate two commercial terminals on the Gaza boundary as border crossings, which could choke off Palestinian trade.
There was no immediate comment on the report from officials close to Mofaz, who is running for leadership of the right-wing Likud party. Israel holds a general election in March.
Saeb Erekat, the chief Palestinian negotiator, said he was unaware of such a threat being made, but said it would be inconsistent with the US-led "road map" to peaceful Palestinian statehood in the West Bank and Gaza, alongside Israel.
"This should not be acceptable to anyone," Erekat said.
Israel withdrew from Gaza in September after 38 years of occupation, and hopes for peace improved. But violence has continued.
In one incident, a Palestinian suicide bomber killed five Israelis on Monday and, in response, Mofaz ordered a resumption of air strikes against militants, in which three have since been killed.
Political sources said Israel has also suspended talks with the Palestinians on allowing bus convoys between Gaza and the West Bank, despite a deal brokered last month by Condoleezza Rice, the US secretary of State.
That agreement led to the Palestinians taking control of the main Gaza-Egypt border crossing terminal at Rafah and also called for the bus convoys to start next week, followed by truck convoys in January, using a "safe passage" to the West Bank.
In addition to Rafah, Palestinians in Gaza have two commercial crossings into Israel, Karni and Erez.